How Do You Measure Diversification? Choueifaty and Coignard Gave It a Number

"Diversification is the only free lunch in investing." Everyone in finance has said this. Now try to answer a simple follow-up: is portfolio A more diversified than portfolio B?

You will find you have no rigorous way to answer. Counting holdings does not work: a portfolio of 500 tech stocks is not diversified. Looking at sector spread is arbitrary. Average correlation is closer, but ignores volatility. The industry's central virtue had no yardstick.

In 2008, Yves Choueifaty and Yves Coignard proposed one, and then did the natural thing: they built the portfolio that maximizes it.

The problem: diversification was a vibe, not a measurement

Without a definition, you cannot optimize for something. You also cannot check whether you have it. Investors habitually assumed that broad market-cap-weighted indices were diversified, because they hold thousands of names. But a cap-weighted index concentrates in whatever has recently become large, which means it is systematically most concentrated in whatever has recently run up. In 1999 the S&P 500 was a technology bet. In 2007 it was heavily financials. Thousands of names, one dominant exposure.

So the industry's default "diversified" portfolio was arguably a momentum-chasing concentration machine. That is the gap the paper attacks.

The key idea via analogy: is the whole calmer than its parts?

Here is the intuition. Suppose you own a basket of assets. Each individual asset has its own volatility. Now imagine, hypothetically, that all your assets moved in perfect lockstep. Then the portfolio's volatility would just be the weighted average of the individual volatilities. There would be no diversification benefit at all, because nothing offsets anything.

In reality, assets do not move in lockstep, so the portfolio's volatility comes out lower than that weighted average. How much lower? That gap is precisely the benefit diversification bought you.

So Choueifaty and Coignard define the diversification ratio:

the weighted average of the individual asset volatilities, divided by the volatility of the portfolio as a whole.

If the ratio is 1, you got nothing: your assets are effectively one asset. If the ratio is 3, your portfolio is a third as volatile as the sum of its risks would suggest. Higher ratio, more diversification. Simple, and it captures both ingredients that matter: how volatile the parts are, and how much they cancel each other out.

The Most Diversified Portfolio is then just the long-only portfolio with the highest possible diversification ratio.

There is a rather elegant theoretical hook here. If you were to assume that every asset's excess return is proportional to its volatility, meaning all assets have the same Sharpe ratio, then the portfolio that maximizes the diversification ratio is the portfolio that maximizes the Sharpe ratio. So the Most Diversified Portfolio is the maximum-Sharpe portfolio under a particular, deliberately agnostic assumption about returns: rather than pretending to forecast which assets will do best, you assume they are all equally rewarding per unit of risk and let the correlation structure do all the work.

That is a much more honest posture than pretending you know expected returns, and it is the philosophical core of all risk-based investing.

What they found

Applied to real equity universes, the Most Diversified Portfolio produced results that were, in the authors' tests, favorable relative to cap-weighted benchmarks, equal-weighted portfolios, and minimum-variance portfolios on a risk-adjusted basis. It also produced portfolios that behaved sensibly: it does not simply pile into the calmest stocks the way minimum variance can, because a stock with high volatility that is uncorrelated with everything else is genuinely useful to a diversification-maximizer, and gets a real weight.

That is the practical distinction worth remembering. Minimum variance loves calm assets. Maximum diversification loves uncorrelated assets. A wild but independent asset is a liability to the first and an asset to the second.

Why it mattered

• It made diversification measurable. Whatever you think of the resulting portfolio, the diversification ratio is a genuinely useful diagnostic. You can compute it for any portfolio you already hold and get a single number for how much your holdings actually offset each other. Applied to cap-weighted indices at various points in history, it makes the concentration problem visible.
• It joined the risk-based investing family. Alongside minimum variance, equal risk contribution and equal weighting, the Most Diversified Portfolio is part of the group of methods that build portfolios without forecasting returns. That family exists because expected returns are the most error-prone input in finance, and every one of these methods is a different answer to the question: if I refuse to guess returns, what should I do instead?
• It became a real product. The authors founded an asset manager around the approach. Whatever the academic debates, the concept was implemented at scale, which is a form of validation that not many portfolio construction papers get.
• It reframed the index question. The paper is part of the broader argument that a cap-weighted index is not a neutral default. It is an active choice that systematically overweights whatever is currently expensive and popular.

The honest limitations

• It is a covariance matrix bet, like all its cousins. The diversification ratio is computed entirely from volatilities and correlations. Both are estimated with error, both are unstable, and correlations in particular tend to jump toward one in a crisis. A portfolio maximized for diversification under normal-period correlations may find that its diversification evaporates exactly when it is needed.
• The equal-Sharpe assumption is a strong one. The elegant "maximum diversification equals maximum Sharpe" result only holds if every asset offers the same return per unit of volatility. That is a deliberately agnostic assumption, but it is still an assumption, and it is false in any world where some assets genuinely carry a risk premium and others do not.
• It can load up on odd corners. Because the optimizer rewards assets that are uncorrelated with the rest, it can develop enthusiasm for obscure or illiquid names whose low measured correlation is partly an artifact of stale prices rather than genuine independence. This is a well-known trap in any correlation-driven optimization.
• The empirical case is from the people selling it. The original results come from the authors, who founded a firm on the method. That does not make them wrong, but the usual caution about backtests produced by those with an interest in the outcome applies, and independent replications are the right thing to look for.

The one-line takeaway

Choueifaty and Coignard turned the vague virtue of diversification into a specific ratio, the weighted average risk of your holdings divided by the risk of your portfolio, and then built the long-only portfolio that maximizes it, giving risk-based investing both a yardstick and a strategy.